singlepixelcompressivespectralpolarization ... ·...

Revista Facultad de Ingeniería, Universidad de Antioquia, No.88, pp. 91-99, 2018

Single pixel compressive spectral polarizationimaging using amovablemicro-polarizer arrayMuestreo compresivo de imágenes espectro polarizadas usando una arquitectura óptica deúnico píxel y una matriz de micropolarizadores móvil

Jorge Luis Bacca-Quintero 1*, Héctor Miguel Vargas-García 2, Daniel Ricardo Molina-Velasco 3, Henry Arguello-Fuentes 1

1Escuela de Ingeniería de Sistemas e Informática, Universidad Industrial de Santander, Carrera 27 # 9. C. P. 680002 Bucaramanga,Colombia2Escuela de Ingeniería Eléctrica, Electrónica y Telecomunicaciones, Universidad Industrial de Santander, Carrera 27 # 9. C. P. 680002Bucaramanga, Colombia3Escuela de Química, Universidad Industrial de Santander, Carrera 27 # 9. C. P. 680002 Bucaramanga, Colombia.

ARTICLE INFO:Received February 20, 2018Accepted August 29, 2018

KEYWORDS:Stokes parameters,Compressive sensing,Spectral polarizationimages, Imagesreconstruction.

Parametros de Stokes,Muestreo compresivo,Imágenes espectropolarizadas,Reconstrucción deimágenes.

ABSTRACT: The acquisition of spectral polarization images is amethod that obtains polarized,spectral and spatial information of a scene. Traditional acquisition methods use dynamicelements that capture all the information of a scene, by scanning the areas of interest, whichresult in large amounts of data proportional to the desired image resolution. Hence, in thiswork, the compression of spectral polarization images using a single pixel architecture,that uses a micro-polarizer array aligned with a binary coded aperture is proposed. Themicro-polarizer is moved horizontally in each shot, so that diverse types of codificationsfrom the scene are obtained. The proposed architecture allows several compressive 2Dprojections with spatial, spectral and polarization coding to be obtained. Stokes parameterimages at several wavelengths are reconstructed. This architecture reduces the totalnumber of measurements needed to obtain spectral polarization images compared totraditional acquisition methods. The experiments validate the quality of the architectureobtaining 43.19 dB, 37.49 dB and 30.41 dB of the peak signal-to-noise ratio for the first threeStokes parameters respectively.

RESUMEN: La adquisición de imágenes espectro polarizadas es un método que obtieneinformación espacial, espectral y de polarización de una escena. Los métodos tradicionalesde adquisición utilizan elementos dinámicos que capturan la totalidad de la informaciónde la escena, escaneando las áreas de interés. Esto resulta en grandes cantidades dedatos proporcionales a la resolución de imagen deseada. Por esta razón, en este trabajo sepropone la compresión de imágenes espectro polarizadas usando una arquitectura ópticade único píxel, que usa un arreglo de micro-polarizadores alineados con una aperturacodificada binaria. El arreglo de micro-polarizadores es movido horizontalmente en cadacaptura, permitiendo lograr diversos tipos de codificaciones de la escena. La arquitecturapropuesta permite obtener varias proyecciones 2-D comprimidas con codificación espacial,espectral y de polarización, para luego reconstruir los Parámetros de Stokes a variaslongitudes de onda. Esta arquitectura reduce el numero total de medidas necesariaspara obtener la información espectral y de polarización de las imágenes comparadas conlas arquitecturas tradicionales. Los experimentos validad la calidad de la arquitecturapropuesta obteniendo 43.19 dB, 37.49 dB y 30.41 dB de la proporción máxima de señal aruido para los tres primeros de Stokes respectivamente.

1. Introduction

The acquisition of spectral images allows obtaininginformation from different ranges of the electromagnetic

spectrum of a scene. The information can be representedas a data cube composed of different images at aspecific wavelength, where each spectral band, providesinformation about the physical properties and distributionsof materials in the scene [1]. On the other hand, one of thephysical quantities associated with nature is polarization[2]. It measures information about the vector nature of theoptical field in the scene, which allows knowing properties

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* Corresponding author: Jorge Luis Bacca-Quintero

E-mail: [email protected]

ISSN 0120-6230

e-ISSN 2422-2844

DOI: 10.17533/udea.redin.n88a10

J. L. Bacca-Quintero et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 88, pp. 91-99, 2018

of the object surface such as roughness, shape, shadingand orientation [3, 4].

Spectral polarization images arise from the union of thesetwo types of images, spectral and polarized, thus obtainingmore information of the scene. Therefore, they have beenused in diverse applications such as classification ofvegetation [3], identification of surfaces contaminatedwith chemical agents [5], and biomedical diagnosis inthe analysis of skin [6]. The major difficulty with spectralpolarization images is their acquisition, because it needsto sense the spatial, spectral and polarization information,F ∈ RM×N×L×θ (see Figure 1). For instance, an imagewithM = N = 256 , L = 14 and θ = 4 demands sensingand storing more than 3 million voxels. Traditionalacquisition methods use a linear polarizer that is rotatedwhile sequential measurements are captured, spanningthe scene in each dimension [7] or by changing the setof color filters [8]. The time required for these methodsdepends directly on the speed of change of these opticalelements, therefore limiting their usage in dynamicsensing while the noise increases due to the changingmechanisms.

On the other hand, compressive sensing devices usedin spectral polarization imaging obtain compressedprojections of a scene, where the number of samplesis smaller than the total amount of scene voxels,which enable faster acquisitions. For instance, recentworks have shown good results with as few as 20% ofcompressed measurements [9–11] which would reduceby 80% the acquisition time of traditional methods. Thetechniques of compressive sensing imaging (CSI) areable to reconstruct the image from an underdeterminedsystem of linear equation that describes the measurementacquisition process [12, 13], by choosing an appropriaterepresentation basis where the image presents sparsebehavior [14].

A single-pixel polarimetric imaging spectrometer wasproposed recently, enabling the acquisition of spatial,spectral, and polarization information about the scenefrom compressive measurements [10]. This architectureutilizes a Digital Micromirror Device (DMD) as a spatiallight modulator. The spectral polarization analysisis achieved by combining a rotating polarizer withthe spectrometer. However, compression occurs inthe spatial domain, while spectral and polarizationdimensions are preserved. Consequently, hundreds ofsequential measurements are needed to obtain a goodconstruction.

Another compressive spectral polarization imagingtechnique that uses a pixelized polarizer and coloredpatterned detector (CSPI) was proposed in [9], this

Figure 1 4-D representation of a spectral polarization images

architecture employs a pixelized polarizer and coloredpatterned detector that enables compressive sensing overspatial, spectral, and polarization domains. However,this architecture only allows four different acquisitionsof the scene and its associated cost increases as eithermore filters are added to the detector or the sensorresolution increases. This limits the use of this system.Therefore, achieving a high quality reconstruction with alow-resolution camera is desired.

For this reason, this paper proposes an alternative toreduce the acquisition cost of the spectral polarizationimages, since it uses a single pixel as the detector.In addition, this architecture can capture multipleshots, using a movable pixelized polarizer and thebinary coded aperture. In this way, obtaining spectraland polarization information of the scene from fewcompressed spatial-spectral and polarization informationmeasurements.

2. Spectral Polarization Images

Spectral polarization images can be modeled as a 4Dstructure, shown in Figure 1, where each 3D imagerepresents the scene at one of the four polarization angles(0◦, 45◦, 90◦, 135◦). This representation does not modifythe spatial structure of the scene. One of the mostcommon ways to represent polarization is by means of theStokes parameters S. These are four vectors that describepartial or total polarization of light based on intensitymeasurements [15]. Stokes vectors are defined in termsof optical intensity as follows: S0 is the total intensity ofa scene, S1 is the difference between the intensity alongthe x (0◦) axis, and the one oriented parallel to the y (90◦)axis, S2 is the difference between the linear +45◦ and−45◦ polarization and S3 is the difference between theintensity transmitted by a right circular polarizer and aleft circular polarizer [16]. In the majority of applications,the S3 component is not used; additionally, sensing S3

requires an additional quarter-wave plate [17], which isnot considered in this work. For this reason, it is typicalto work with only the first three Stokes vectors, whichhave a linear relationship with the measurements of thetraditional detectors. These are given in Equations 1,2 and

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3 as follows

S0 = I0 + I90 = I45 + I135 (1)

S1 = I0 − I90 (2)

S2 = I45 − I135 (3)

where Iθ is the polarization intensity at the angle θ, S0

is the total radiation of a beam, and S1 and S2 are theradiation difference of the linearly polarized beam, theseparameters are visualized in Figure 2.

Figure 2 Visual representation of the Stokes parameters inblack and white images. a) S0 , b) S1 and c) S2 at a wavelength

of 530 nm

The angle of polarization (AoP ) ), which is defined byEquation. 4, specifies the orientation of the beamoscillation [9], which in terms of the Stokes parameters canbe represented by

AoP =1

2arctan

(S2

S1

). (4)

As we can see, the angle depends only on the parametersS1 and S2.

3. Sampling Process

To capture spectral polarization images in a compressedmanner, we propose the optical architecture shown inFigure 3. There, the scene is encoded in polarizationand in spectrum by the pixelated polarizer and the codedaperture. Then the coded scene passes through thecondenser lens, which concentrates the light to a point,creating a mixed pixel, which contains all the encodedinformation. This point is integrated by the spectrometerthat divides the information into spectral ranges.

In the sampling scheme, the scene is represented byf(x, y, λ, θ), where x and y represent the two spatialdimensions, λ is the spectral wavelength, and θ representsthe angle of linear polarization. By considering thepossibility of applying multiple measurements, the scenepasses through a polarization filter array u(k)(x, y, θ)|k =1, . . . ,K, that allows or not the pass of a certainpolarization angle per pixel, with K possible patterns.Then it encounters a coded aperture c(k)(x, y)|k =

Figure 3 Scheme of the proposed single pixel camera forspectral polarization data acquisition

1, . . . ,K, which applies spatial modulations to the scene.Ideally, u and c are binary functions, the blocks blocking ornot the passing of voxels in the 4D data cube. In this way,the spatial, spectral and polarization modulated scene isobtained as in Equation. 5

f(k)1 (x, y, λ, θ) = u(k)(x, y, θ)c(k)(x, y)f(x, y, λ, θ). (5)

Let u(k)(i,j,r), c

(k)(i,j) and f(i,j,l,r) be the discretized

polarization filter array, the discretized coded apertureand the discretized data respectively defined in Equation.6,7 and 8 as

u(k)(x, y, θ) =∑i,j,r

u(k)(i,j,r)rect(x, y)δ(θ) (6)

c(k)(x, y) =∑i,j

c(k)(i,j)rect(x, y) (7)

f(x, y, λ, θ) =∑i,j,r

f(i,j,r)(λ)rect(x, y)δ(θ) (8)

where

rect(x, y) =

1(i− (1/2))∆ ≤ x < (i+ (1/2))∆

(j − (1/2))∆ ≤ y < (j + (1/2))∆

0 otherwise

δ(θ) =

{1 (r − 1)∆θ ≤ θ < (r + 1)∆θ

0 otherwise

are the 2D and 1D sampling functions respectively, ∆is the sample pixel size which is assumed equal for themicro-polarizer array, the coded aperture and the imagesand∆θ is the sample angle. The discrete form describingthe modulation of the scene given in Equation. 5 isexpressed in (9) as:

f(k)1 (x, y, λ, θ) =

∑i,j,r

u(k)(i,j,r)c

(k)(i,j)f(i,j,r)(λ)×

rect(x, y)δ(θ)

, (9)

and the continuous model for the spectral density throughthe coded aperture, the polarization filter array and theoptics before it impinges the sensor array is given by

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Equation.10

f(k)2 (λ) =

∫∫∫f(k)1 (x, y, λ, θ)dxdydθ

=∑i,j,r

u(k)(i,j,r)c

(k)(i,j,)f(i,j,r)(λ).

(10)

In the proposed sensing model, the scene is viewed as fourlinear polarization intensity cubes indexed by r = 1, 2, 3and 4 indicating cubes with four polarization angles and∆θ = 45. Also, the spectral range of the instrumentis partitioned into a finite number of subintervals orchannels. The discretization of the spectral axis is givenas λ(l) for l = 1, ..., L where L is the number of spectralbands. The range of the channel l is [λ(l), λ(l+1)] whereλ(l) is the solution of the Equation. 11

S(λ(l+1))− S(λ(l)) = ∆, l = 1, ..., L (11)

where this pixel is taken by the spectrometer to obtainmeasurements by spectral bands in (12) as

y(k)(l) =

∫f(k)2 (λ)δ(λ)dλ+ ω

(k)(l) , (12)

where

δ(λ) =

{1 (l − (1/2))∆λ(l) ≤ θ < (l + (1/2))∆λ(l)

0 otherwise,

ω(k)(l) is additive noise in the sensor and ∆λ(l) = λ(l+1) −

λ(l), l = 1, ..., L is the range of the spectral band l. Finally,in Equation. 13 and 14 the discrete model to obtain themeasurements is given as

y(k)(l) =

4∑r=1

N∑i,j=1

u(k)(i,j,r)c

(k)(i,j)f(i,j,r,l)

+ ω(k)(l) , (13)

with

f(i,j,r,l) =

∫f(i,j,r)(λ)δ(λ)dλ (14)

is the discretized data and N is the spatial resolution.By converting from row-column subscripts into linearindexing as n = N(i − 1) + j, for i, j = 1, ..., N , theEquation. 13 becomes (15) as

y(k)(l) =

4∑r=1

N2∑n=1

u(k)(n,r)c

(k)(n)f(n,r,l)

+ ω(k)(l) , (15)

and the matrix form is expressed in Equation.16 as

y(l) =[H(1), . . . ,H(4)]

f(1,l)...f(4,l)

+ ω(l)

=Hf(l) + ω(l)

, (16)

where f(r,l) = [f(1,r,l), . . . , f(N2,r,l)] is the vectorizationof the spectral polarization imaging in the angle r andband l, y(l) = [y(1,l), . . . , y(K,l)] are the compressive

measurements in the band l and H(r) ∈ RK×N2

is thesampling matrix which is determined by the polarized andcoded aperture in Equation.17 as

H(r) =

u(1)(1,r)c

(1)(1) . . . u

(1)(N2,r)c

(1)(N2)

.... . .

...u(K)(1,r)c

(K)(1) . . . u

(K)(N2,r)c

(K)(N2)

. (17)

Because all spectral bands are encoded with the samecoded aperture pattern, in Equation. 18 the problem canbe seen in a vector way as

y = [H(1), H(2), H(3), H(4)][fT(1), f

T(2), f

T(3), f

T(4)]

T = Hf + ω(18)

where y ∈ RK×L are the compressive measurementsobtained by the spectrometer in K shots, H(r) = I(L) ⊗H(r), where I(L) is an identity matrix of size L, ⊗ denotedthe Kronecker product, and f(r) are the vector images atthe angle r.

3.1 Measurement hardware strategy

The vector form of the coded aperture is given by c(i) =[c(1), c(2), ..., c(N2)]

T for k = 0, 1, ...,K. Expressing theset of coded apertures and considering K total shots inEquation.19 we have

C = [c(1), c(2), ..., c(K)]T , (19)

whereC ∈ {0, 1}K×N2

represents the binary value (whitetranslucent or block). Designing an array of polarizers thatchanges at each acquisition is expensive [9], this paperproposes designing one with dimensionsN×(N+K−1),such that for each capture, the array of polarizers is movedhorizontally in a pixel. Mathematically,U can been seen asa 3D array of binary elements that represent the pixelatedpolarizer, (see Figures 4(a) and 4(b)) in which an angle ris represented as u(r) = [u(r,1), u(r,2), ..., u(r,S)]

T whereu(r) ∈ {0, 1}S and S = N×(N+K−1) for r = 1, 2, 3, 4.Then, in Equation.20 for each shot we have

u(r,k) = [u(r,(k−1)N+1), u(r,(k−1)N+2), ..., u((r,(k−1)N+N2)]T ,(20)

for k = 0, 1, ...,K which represents the horizontalmovement of a pixel for the angle r, this can be seenin Figure 4(c) for the first vector of u1,1. The binarymatrix is expressed in Equation.21, which represents allthe acquisitions

U(r) = [u(r,1), u(r,2), ..., u(c,K)]T (21)

whereU(r) ∈ {0, 1}K×N2

for r = 1, ..., 4.The sampling matrix H(r) which is determined usingEquation.22, represents the sampling, modulation and the

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Figure 4 Representation of the pixelated polarizer a) visualrepresentation b) binary representation c) vectorization of the

first angle and first shot of the binary representation

Figure 5 Illustrative example of the sensing matrix H forN = 4,M = 4, L = 3. White points have the values of 1 and

black points are 0

different captures of the spectral polarization images. Theinformation that thismatrix has is shown in a specific orderas:

H(r) =

C ◦U(r) 0 · · · 0

0 C ◦U(r) · · · 0...

.... . .

...0 0 · · · C ◦U(r)

,

(22)where C ◦ U(r) is the Hadamard product between thematrices C and U(r). A graphical representation of thesampling matrix is shown in Figure 5, For this example, animage with 4 × 4 pixels of spatial resolution, 3 spectralbands, 4 polarization angles and 50% of compression isused. The compression rate is calculated as γ = S

4MN .The white points represent the unblocking pixel (1), whilethe entries (0) are represented in black.

The relationship between the intensity of the light passingthrough a θ◦ linear polarizer Iθ and the Stokes parameters

S0 to S2, of the original light, is linear and given by thefollowing Equation. 23

Iθ =1

2S0 +

1

2cos(2θ)S1 +

1

2sin(2θ)S2. (23)

Therefore, the vectorized linear polarization cubes f havea linear transformation with the three first three Stokesparameter cubes s, as shown in Equation.24

f = Es, (24)

where E = [ET1 ,E

T2 ,E

T3 ,E

T4 ]

T and Er ∈ RMNL×3MNL

consist of three diagonal block matrices expressed inEquation.25 as:

Ec =

[diag

(1

2

), diag

(1

2cos 2θ(r)

), diag

(1

2sin 2θ(r)

)],

(25)for the four values of θ(r) with r = 1, .., 4. Thus, thesensing process referent to the Stokes parameter can beexpressed, as in Equation.26

y = HEs = Gs+ ω (26)

where G represents the sensing process from the treeStokes parameter cubes directly to the measurements.Due to matrix H having inputs 1 and 0, andE entries givenin Equation. 25, the values of G are given from the set{− 1

2 , 0,12 , 1,

32 , 2}. In order to see the sensing matrix G

an image of 4 × 4 pixels of spatial resolution, 3 spectralbands, 4 polarization angles and 50% of compression isused. The new compression rate with respect to Stokesparameters is calculated as γ = S

3MN , where 3 arerepresenting the three first Stokes parameters. The bluepoints represent− 1

2 , black represents 0, green represents12 , white represents 1, red represents 3

2 and finally yellowrepresents 2. In Figure 6 the sensing matrix can be seen,that represents the first parameter where its minimumvalue is 0 and its maximum value is 2, and for the othertwo parameters the values are − 1

2 , 0,12 , due to the values

taken by the product between H and E.

To exploit the sparsity of the data cube, each Stokesparameter is represented by a three dimensionalKronecker basisΨ = Ψ1 ⊗Ψ2 ⊗Ψ3, whereΨ1 ⊗Ψ2 isthe 2D-Wavelet basis that provides the basis in the spatialdomain and Ψ3 is the discrete Cosine basis that is thebasis in the spectral domain. In this case s = ΨΘ, thusthe sensing process can be expressed in Equation.27 as

y = Gs+ ω = GΨΘ+ ω = AΘ+ ω, (27)

where A is the composite sensing matrix that modulesthe system. The signal recovery is obtained by solving theinverse problem of the under determined linear system in(27). This consists in recoveringΘ such that the l1−l2 cost

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Figure 6 Illustrative example ofG, which is composed of thethree sensing matrices corresponding to the first three Stokesparameters forN = 4,M = 4, L = 3. Blue points represent− 1

2, black points are 0, green represents 1

2, white points have

the values of 1, red represents 32and yellow points are 2

function is minimized [14, 18]. The optimization problem isgiven Equation.28 as

Θ = arg minΘ∥y −AΘ∥22 + λ∥Θ∥1, (28)

where λ is a regularization parameter. The GradientProjection for Sparse Reconstruction (GPSR) algorithm[19] is used to solve Equation. 28 in this work.

4. Design of the sampling matrixbased on Hadamard matrices

Recent work has shown that designing sampling matricessignificantly improves the quality of the reconstruction [20–22]. In this section, the Hadamard matrix is used to designthe sensing matrix since its rows are mutually orthogonal;this property is desired in compressive sensing [12, 23].This property allows a fast reconstruction approach, dueto the transpose normally used in the GPSR algorithmis reduced to only one matrix product [18, 24]. Thus,Equation.29 is

C ◦ U(r) = P1MhP2 (29)

where Mh ∈ {−1, 1}N2×N2

is a Hadamard matrix, P1 ∈{0, 1}K×N2

is an incomplete permutation matrix that onlyhas a one-valued entry on each row andP2 ∈ {0, 1}N2×N2

is a permutation matrix which operates over the columns

ofMh [25]. Therefore, the sensingmatrix is nowexpressedin Equation.30 as

Hc =

C ◦ U(r) 0 · · · 0

0 C ◦ U(r) · · · 0...

.... . .

...0 0 · · · C ◦ U(r)

(30)

In order to apply this codification the entries of H shouldbe {−1, 1} instead of {0, 1}. For this, the measurementsy0 = Df are firstly taken, where D is a sensing matrixwith C(i,j) = 1 and U(r,i,j) = 1,∀i,j,r, letting allthe information of the scene pass in an acquisition. Thecodified measures obtained with {−1, 1} for each shot arecalculated using Equation.31 as

y = 2y − yo = (2H−D)f = Hf , (31)

where H represent a shot of the sensing matrix expressedin (30). The problem with noise is expressed in Equation.32as

y = (2H+ ω1)f − (Df + ω2)

= (2H−D)f + 2ω1 + ω2

= Hf + ω

(32)

where ω is the noise present in the process. The sensingand reconstruction referring to the parameters of stokesare followed from Equation. 26 replacing H by H.

5. Simulations and Results

To evaluate performance and study the proposedcompressive sensing system, simulations were performedwith a 4D test data array, which contains four cubes ofpolarization intensities that were acquired by switchingfourteen bandpass filters combined with four azimuthangles (0◦, 45◦, 90◦ and 135◦) of a linear polarizer(LPVISB100-MP2). Each cube with polarization intensitycontains fourteen (L = 14) spectral bands ranging from(500 nm to 620 nm), with spatial resolution of 256 by256. The scene was illuminated with unpolarized light. InFigure 7 the scene shows the four polarization angles infour different spectral bands.

The linear polarization information is obtained by Eqs.1, 2 y 3. Figure 8 shows the Stokes parameters forfour different wavelength. The second and third Stokesparameters represent the linear polarization state. In the4D array of data, a toy and a bulb can be seen, each withdifferent textures and shapes. With this four-dimensionaldata cube, simulations can be performed using Equation.15 without additional noise.

The proposed architecture was compared with CSPI [9],this was used with random inputs for the micropolarizer

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Figure 7 4-D representation of a spectral polarization images infour spectral channels

and the colored filter array. The GPSR algorithmwas used to reconstruct the Stokes parameters fromcompressed measurements for both architectures. Thepeak signal-to-noise ratio (PSNR) is used to measure thequality of the reconstructed Stokes parameters.

The compression level of CSPI is given as γ = SNm

PMNL ,where P = 3 is the number of Stokes parameters used,S is the number of shots and Nm = (M + L − 1)Nis the number of measurements in a single acquisition.It should be clarified that, for this architecture only 4shots can be made, because the prism is rotated only in 4angles, so for a single shot its compression level is 2.5%and the maximum compression level for this architecturewould be 10% for these images. On the other hand, thecompression level of the proposed architecture is given asγ = S

PMN . In our architecture, the number of differentcaptures depends on the size of the micro-polarizer andbecause there is a coded aperture that may vary with eachshot, the number of encodings other than the scene for asingle movement of the micro-polarizer is given by

(MNMNτ

)where τ =

∑m,n t(m,n)/MN is the quantity of energy

that passes through an object known as transmittance,allowing multiple acquisitions.

Figure 9 shows the average PSNR of 20 iterations fordifferent levels of compression from 5% to 10% with stepof 2,5 and from 10% to 50% with step of 5, for 3 stokesparameters reconstructed. It can be seen that, for 2.5%to 10% levels of compression the proposed architectureoutperforms CSPI in the parameters S1 and S2. In aparticular case, for 2.5% the proposed method overcomes

500 nm 500 nm 500 nm

530 nm 530 nm 530 nm

580 nm 580 nm 580 nm

620 nm 620 nm 620 nm

S0

S0

S0

S0

S1

S1

S1

S1

S2

S2

S 2

S2

Figure 8 Stokes parameter S0, S1 and S2 for each data cube isshown in four of 14 bands of polarization: 500, 530, 580 and 620

nm

up 4.5 dB and 5.3 dB for S1 and S2 respectively, for theparameter S0 both methods have a similar quality. Thedotted line represents the maximum PSNR achieved byCSPI because taking more snapshots is not possible.

To visualize the reconstruction quality, the reconstructedStokes images plane in four spectral channels for 10%of compression are displayed in Figure 10 for botharchitectures. The reconstruction shows significant imagequality compared to CSPI.

In order to verify the spectral accuracy of the proposedarchitecture, three spectral points of the original data cubeare compared with the reconstructed signatures. In Figure11 the results are presented. In general, the results showthat the proposed architecture presents better spectralperformance than CSPI.Finally, to visualize the reconstruction quality with morelevel of compression Figure 12 shows reconstructed Stokesparameters for each data cube in four of 14 bands ofpolarization: 500, 530, 580 and 620 nm. It can be seenthat with 30% of compression the reconstruction has goodimage quality.

6. Conclusion

The mathematical and matrix model for the single-pixelarchitecture for the compressive acquisition of spectral

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Figure 9 Mean PSNR of 20 iteration for different levels ofcompression from 5% to 10% with step of 2,5 and from 10% to50% with step of 5, for a) the first b) the second and c) the third

parameter reconstructed

polarization images was developed. The architecturepresented makes use of a micro-polarizer that allows ordenies the propagation of the polarization angles of theimage, a coded aperture that allows the spectral andspatial coding, the collimator modulates the informationto a pixel and this is classified in spectral bands using thespectrometer. The coding of the scene produced by themicro-polarizer and the coded aperture was analyzed fordifferent levels of compression, the results show a gain ofup to 3dB for 10% compression in the parameters S1 andS2 compared to CSPI architecture, also, 30% compressionexhibited stable quality for the studied image. Future workincludes the implementation of the proposed architectureto validate the obtained results in a real scenario.

Figure 10 Reconstructed Stokes parameter S0, S1 and S2

using the proposed and CSPI architecture. The parameter S0 isdisplayed in the band 530 nm, S1 in the band 600 nm and S2 in

540 nm

Figure 11 a) RGB image of the sample and spectral signaturesobtained in the reconstructions with the proposed and CSPI

architectures, in the points of the image b) P1 (x = 94, y = 175) inthe parameter S0 c) P2 (x = 48 , y = 175) in parameter S1 and d)

P3(x = 137, y = 177) in parameter S2

7. Acknowledgments

The authors would like to thank the support providedby the Vicerrectoría de Investigación y extensión ofUniversidad Industrial de Santander under the VIE code1867, entitled ”Diseño y simulación de un sistemade muestreo compresivo para señales de resonanciamagnética”.

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Figure 12 Reconstructed Stokes parameter S0, S1 and S2 foreach data cube is shown in four of 14 bands of polarization: 500,

530, 580 and 620 nm

References

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